•Loop contribution of fourth order in the chiral expansion

mesonstop = CreateTopologies[1, 1 -> 1, Adjacencies -> {3, 4}] ;

mesontreeinsert = InsertFields[mesonstop, {Pion[0, {i1}]} -> {Pion[0, {i2}]}, Model -> "Automatic", GenericModel -> "Automatic", InsertionLevel -> Classes] ;

Paint[mesontreeinsert, PaintLevel -> {Classes}, AutoEdit -> False, SheetHeader -> False, Numbering -> False, ColumnsXRows -> {2, 1}] ;

[Graphics:../HTMLFiles/index_51.gif]

[Graphics:../HTMLFiles/index_52.gif]

$ConstantIsoIndices = Union[$ConstantIsoIndices, {i1, i2, I1}] ;

amplFC = CreateFCAmp[mesontreeinsert, EqualMasses -> False, Sum -> True] /. i2 -> i1 // SUNReduce[#, FullReduce -> True] & // Simplify

{-1/(96 π^4 (f _ π^(ó  ))^4 (q _ 1^2 - (m _ π^(I _ 1 ))^2)) (i (4 C^( ) (δ _ (3 I _ 1)^(2) - 4 δ _ (i _ 1 I _ 1)^(2) + δ _ (3 i _ 1)^(2) (4 δ _ (3 I _ 1)^(2) δ _ (i _ 1 I _ 1)^(2) + 1) - 2) (e^( ))^2 + (f _ π^(ó  ))^2 (-(2 δ _ (i _ 1 I _ 1)^(2) + 1) (m _ π^0^(ó  ))^2 + 2 p _ 1 · p _ 3 (δ _ (i _ 1 I _ 1)^(2) - 1) - 2 q _ 1^2 (δ _ (i _ 1 I _ 1)^(2) - 1))) SumOver(I _ 1, 3)), (i (e^( ))^2 g^(μ _ 1 μ _ 2)^2 (δ _ (3 i _ 1)^(2) - 1))/(16 π^4 (q _ 1^2 - (m _ γ^(ó  ))^2)), 0, -(i (e^( ))^2 g^(μ _ 1 μ _ 2) (p _ 1^μ _ 1 + q _ 1^μ _ 1) (p _ 3^μ _ 2 - q _ 1^μ _ 2) (f _ (3 i _ 1 I _ 1)^(2))^2 SumOver(I _ 1, 3))/(16 π^4 (q _ 1^2 - (m _ π^(I _ 1 ))^2) . ((p _ 3 + q _ 1)^2 - (m _ γ^(ó  ))^2))}

ampreduced = (OneLoop[q1, #, Sum -> Explicit]) & /@ amplFC ;

ampsimple = ampreduced // IsoToChargedMasses // SUNReduce // Simplify

{-1/(96 π^2 (f _ π^(ó  ))^4) (2 A _ 0 ( (m _ π^+^(ó  ))^2 ) (4 C^( ) (2 δ _ (1 i _ 1)^(2) + 2 δ _ (2 i _ 1)^(2) - δ _ (3 i _ 1)^(2) + 2) (e^( ))^2 + (f _ π^(ó  ))^2 ((δ _ (1 i _ 1)^(2) + δ _ (2 i _ 1)^(2) - 2) (m _ π^+^(ó  ))^2 - p _ 1 · p _ 3 (δ _ (1 i _ 1)^(2) + δ _ (2 i _ 1)^(2) - 2) + (m _ π^0^(ó  ))^2 (δ _ (1 i _ 1)^(2) + δ _ (2 i _ 1)^(2) + 1))) + A _ 0 ( (m _ π^0^(ó  ))^2 ) (-4 C^( ) (δ _ (3 i _ 1)^(2) - 1) (e^( ))^2 - (f _ π^(ó  ))^2 ((1 - 4 δ _ (3 i _ 1)^(2)) (m _ π^0^(ó  ))^2 + 2 p _ 1 · p _ 3 (δ _ (3 i _ 1)^(2) - 1)))), -((e^( ))^2 (2 A _ 0 ( (m _ γ^(ó  ))^2 ) - (m _ γ^(ó  ))^2) (δ _ (3 i _ 1)^(2) - 1))/(8 π^2), 0, 1/(32 π^2 p _ 3^2) ((e^( ))^2 (-B _ 0 (p _ 3^2, (m _ π^+^(ó  ))^2, (m _ γ^(ó  ))^2) p _ 3^4 - 3 B _ 0 (p _ 3^2, (m _ π^+^(ó  ))^2, (m _ γ^(ó  ))^2) (m _ π^+^(ó  ))^2 p _ 3^2 + B _ 0 (p _ 3^2, (m _ π^+^(ó  ))^2, (m _ γ^(ó  ))^2) (m _ γ^(ó  ))^2 p _ 3^2 + A _ 0 ( (m _ π^+^(ó  ))^2 ) p _ 3^2 - 3 A _ 0 ( (m _ γ^(ó  ))^2 ) p _ 3^2 + 3 B _ 0 (p _ 3^2, (m _ π^+^(ó  ))^2, (m _ γ^(ó  ))^2) p _ 1 · p _ 3 p _ 3^2 - B _ 0 (0, (m _ π^+^(ó  ))^2, (m _ γ^(ó  ))^2) p _ 1 · p _ 3 (m _ π^+^(ó  ))^2 + B _ 0 (p _ 3^2, (m _ π^+^(ó  ))^2, (m _ γ^(ó  ))^2) p _ 1 · p _ 3 (m _ π^+^(ó  ))^2 + B _ 0 (0, (m _ π^+^(ó  ))^2, (m _ γ^(ó  ))^2) p _ 1 · p _ 3 (m _ γ^(ó  ))^2 - B _ 0 (p _ 3^2, (m _ π^+^(ó  ))^2, (m _ γ^(ó  ))^2) p _ 1 · p _ 3 (m _ γ^(ó  ))^2) ((δ _ (1 i _ 1)^(2))^2 + (δ _ (2 i _ 1)^(2))^2))}

ampinfinities = VeltmanExpand[ampsimple, ExplicitLeutwylerJ0 -> True] // Simplify ;

Converted by Mathematica  (July 10, 2003)